September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is an essential subject for anyone who desires to learn arithmetic or another subject that uses it. It's hard work, but we'll ensure you get a handle on these concepts so you can make the grade!

Don’t feel dispirited if you don’t recall or don’t understand these concepts, as this blog will help you understand all the fundamentals. Moreover, we will teach you the secret to learning faster and enhancing your scores in math and other popular subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines meet, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Additionally, the most essential thing to keep in mind is that they are the same in measurement! This means that regardless of where these straight lines cross, the angles converse each other will always share the equal value. These angles are called congruent angles.

Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to discover the others using proportions.

Proving the Theorem

Proving this theorem is somewhat simple. Primarily, let's draw a line and call it line l. Then, we will draw another line that intersects line l at some point. We will name this second line m.

After drawing these two lines, we will name the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:

We understand that angles A and B are vertically contrary because they share the same vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A equals angle B.

If you observe angles B and C, you will notice that they are not linked at their vertex but next to each other. They have in common a side and a vertex, signifying they are supplementary angles, so the sum of both angles will be 180 degrees. This instance repeats itself with angles A and C so that we can summarize this in the following manner:

∠B+∠C=180 and ∠A+∠C=180

Since both additions equal the same, we can sum up these operations as follows:


By removing C on both sides of the equation, we will end with:


So, we can conclude that vertically opposite angles are congruent, as they have identical measurement.

Vertically Opposite Angles

Now that we know the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.


As we mentioned, vertically opposite angles are two angles made by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.

However, vertically opposite angles are never adjacent. Adjacent angles are two angles that have a common side and a common vertex. Vertically opposite angles never share a side. When angles share a side, these adjacent angles could be complementary or supplementary.

In the case of complementary angles, the addition of two adjacent angles will equal 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used to prove the vertical angle theorem.

These theories are appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the properties of vertically opposite angles.

There are several characteristics of vertically opposite angles. Regardless, chances are that you will only require these two to nail your examination.

  1. Vertically opposite angles are always congruent. Hence, if angles A and B are vertically opposite, they will measure the same.

  2. Vertically opposite angles are at no time adjacent. They can share, at most, a vertex.

Where Can You Use Opposite Angles in Real-World Circumstances?

You may speculate where you can find these concepts in the real world, and you'd be surprised to notice that vertically opposite angles are very common! You can locate them in various daily things and scenarios.

For instance, vertically opposite angles are made when two straight lines cross. Back of your room, the door attached to the door frame creates vertically opposite angles with the wall.

Open a pair of scissors to create two intersecting lines and alter the size of the angles. Track junctions are also a great example of vertically opposite angles.

Eventually, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.

Be sure to watch your surroundings, as you will find an example next to you.

Puttingit All Together

So, to sum up what we have considered so far, vertically opposite angles are made from two intersecting lines. The two angles that are not adjacent have the same measure.

The vertical angle theorem states that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line intersecting it and applying the concepts of congruent angles to complete measures.

Congruent angles means two angles that measure the same.

When two angles share a side and a vertex, they can’t be vertically opposite. However, they are complementary if the sum of these angles equals 90°. If the addition of both angles equals 180°, they are deemed supplementary.

The sum of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will at all time add up to 180°.

Vertically opposite angles are quite common! You can discover them in various daily objects and scenarios, such as windows, doors, paintings, and trees.

Additional Study

Search for a vertically opposite angles questionnaire on the internet for examples and exercises to practice. Math is not a onlooker sport; keep practicing until these concepts are well-established in your head.

However, there is nothing humiliating if you require extra assistance. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill instructor can assist you grasp the topic and ace your following test.